Proyecto Parcial 1: Regresión¶
Para este proyecto se utilizó una base de datos llamada Superconductivity Data obtenida del UCI Machine Learning Repository.
La base de diferentes datos y mediciones sobre 21263 materiales superconductores distintos. En total, en la base hay 81 variables de las cuales todas son de tipo numérico, algunos enteros y otros continuos.
La variable de interés es la Temperatura Crítica (critical_temp) ya que esta es una característica muy importante en la aplicación de superconductores en la ingeniería, y poder predecir cual sera la temperatura crítica de un superconductor puede contribuir a la integración de superconductores en nuevas tecnologías.
In [11]:
import pandas as pd
df = pd.read_csv("train.csv")
print("Número de Observaciones:",df.shape[0])
print("Número de Variables:",df.shape[1]-1)
print("Variable de interes:",df.columns.tolist()[81])
print("\nTodas las Variables:")
for i in df.columns:
print("-",i)
Número de Observaciones: 21263 Número de Variables: 81 Variable de interes: critical_temp Todas las Variables: - number_of_elements - mean_atomic_mass - wtd_mean_atomic_mass - gmean_atomic_mass - wtd_gmean_atomic_mass - entropy_atomic_mass - wtd_entropy_atomic_mass - range_atomic_mass - wtd_range_atomic_mass - std_atomic_mass - wtd_std_atomic_mass - mean_fie - wtd_mean_fie - gmean_fie - wtd_gmean_fie - entropy_fie - wtd_entropy_fie - range_fie - wtd_range_fie - std_fie - wtd_std_fie - mean_atomic_radius - wtd_mean_atomic_radius - gmean_atomic_radius - wtd_gmean_atomic_radius - entropy_atomic_radius - wtd_entropy_atomic_radius - range_atomic_radius - wtd_range_atomic_radius - std_atomic_radius - wtd_std_atomic_radius - mean_Density - wtd_mean_Density - gmean_Density - wtd_gmean_Density - entropy_Density - wtd_entropy_Density - range_Density - wtd_range_Density - std_Density - wtd_std_Density - mean_ElectronAffinity - wtd_mean_ElectronAffinity - gmean_ElectronAffinity - wtd_gmean_ElectronAffinity - entropy_ElectronAffinity - wtd_entropy_ElectronAffinity - range_ElectronAffinity - wtd_range_ElectronAffinity - std_ElectronAffinity - wtd_std_ElectronAffinity - mean_FusionHeat - wtd_mean_FusionHeat - gmean_FusionHeat - wtd_gmean_FusionHeat - entropy_FusionHeat - wtd_entropy_FusionHeat - range_FusionHeat - wtd_range_FusionHeat - std_FusionHeat - wtd_std_FusionHeat - mean_ThermalConductivity - wtd_mean_ThermalConductivity - gmean_ThermalConductivity - wtd_gmean_ThermalConductivity - entropy_ThermalConductivity - wtd_entropy_ThermalConductivity - range_ThermalConductivity - wtd_range_ThermalConductivity - std_ThermalConductivity - wtd_std_ThermalConductivity - mean_Valence - wtd_mean_Valence - gmean_Valence - wtd_gmean_Valence - entropy_Valence - wtd_entropy_Valence - range_Valence - wtd_range_Valence - std_Valence - wtd_std_Valence - critical_temp
Huecos¶
La base de datos cuenta con algunos huecos los cuales fueron detectados al momento de intentar generar un modelo de regresión lineal con los datos sin procesar, a manera de prueba. Para resolver esto, se utiliza la funcion dropna() de pandas y se elimina toda aquella columna con huecos NaN.
In [25]:
df = df.dropna()
15667
Outliers¶
Para identificar valores atípicos, se utiliza el método de Tukey y se calcula el rango intercuartíl de todas y cada una de las variables y se eliminan las columnas que tengan valores fuera del rango. Tras esta acción, se reduce el número de observaciones.
In [13]:
import numpy as np
df_filtrado = df.copy()
for column in df.columns:
Q1 = df[column].quantile(0.25)
Q3 = df[column].quantile(0.75)
IQR = Q3-Q1
lower_bound = Q1 - 3 * IQR
upper_bound = Q3 + 3 * IQR
df_filtrado = df_filtrado[(df_filtrado[column] >= lower_bound) & (df_filtrado[column] <= upper_bound)]
print("Nuevo número de observaciones:",df_filtrado.shape[0])
df= df_filtrado
Nuevo número de observaciones: 15667
Colinearidad¶
En esta base de datos, muchas de las variables son medias u otros estadísticos de una misma variable, por lo cual pueden provocar colinearidad. Para identificar si existe este problema se genera un heatmap donde se muestre visualmente, una matriz de correlaciones entre las variables.
In [14]:
import seaborn as sns
import matplotlib.pyplot as plt
y = df['critical_temp']
x = df.drop(columns = ['critical_temp'])
corr_matrix = x.corr()
plt.figure(figsize=(100,100))
sns.heatmap(corr_matrix, annot=True, cmap="coolwarm",fmt=".2f",linewidths=0.5)
plt.title("Correlation Matrix Heatmap")
plt.show()
Como era esperado, hay una gran cantidad de variables que muestran colinealidad alta, para erradicar esto, de cada par de variables con correlación mayor o igual a 0.9, se eliminará una de ellas. Tras esto se reduce el número de variables.
In [15]:
column_purge = set()
for i in range(len(corr_matrix.columns)):
for j in range(i + 1, len(corr_matrix.columns)):
if abs(corr_matrix.iloc[i, j]) >= 0.9:
colname = corr_matrix.columns[j]
column_purge.add(colname)
x = x.drop(columns = column_purge)
print(f"columnas eliminadas por alta correlación: {column_purge}")
corr_matrix = x.corr()
plt.figure(figsize=(25,25))
sns.heatmap(corr_matrix, annot=True, cmap="coolwarm",fmt=".2f",linewidths=0.5)
plt.title("Correlation Matrix Heatmap")
plt.show()
print("Nuevo número de variables:",x.shape[1])
columnas eliminadas por alta correlación: {'wtd_std_ElectronAffinity', 'wtd_entropy_Valence', 'wtd_std_atomic_radius', 'std_Density', 'wtd_gmean_Density', 'wtd_gmean_atomic_mass', 'gmean_fie', 'std_atomic_radius', 'entropy_atomic_mass', 'gmean_Density', 'std_ThermalConductivity', 'wtd_entropy_atomic_radius', 'wtd_gmean_fie', 'wtd_std_Density', 'entropy_Valence', 'wtd_std_ThermalConductivity', 'wtd_mean_atomic_radius', 'wtd_std_atomic_mass', 'std_fie', 'gmean_FusionHeat', 'gmean_Valence', 'entropy_Density', 'range_atomic_radius', 'entropy_FusionHeat', 'wtd_gmean_FusionHeat', 'wtd_std_fie', 'wtd_std_FusionHeat', 'std_Valence', 'std_atomic_mass', 'wtd_gmean_Valence', 'wtd_range_ThermalConductivity', 'entropy_atomic_radius', 'wtd_mean_Valence', 'std_FusionHeat', 'wtd_gmean_atomic_radius', 'entropy_fie', 'std_ElectronAffinity', 'wtd_mean_FusionHeat'}
Nuevo número de variables: 43
División de Datos¶
Para comenzar a entrenar modelos se debe dividir la base de datos en 2. Para esto se usa la función train_test_split ya que el siguiente paso el realizar una selección de características donde se requiere generar varios modelos de regresión.
In [16]:
from sklearn.model_selection import train_test_split
x_train, x_test, y_train, y_test = train_test_split(x,y,test_size = 0.3, random_state = 23)
Selección de Características¶
Para seleccionar las características que serán utilizadas en los modelos de regresión, se utilizará un modelo de selección hacia adelante.
In [17]:
from sklearn.linear_model import LinearRegression
from mlxtend.feature_selection import SequentialFeatureSelector as SFS
from sklearn.metrics import r2_score
model = LinearRegression()
model.fit(x_train, y_train)
sfs_f = SFS(model, k_features = 'best', forward = True, scoring = 'r2', cv = 10, verbose = 1)
sfs_f.fit(x_train, y_train)
selected_features_f = list(sfs_f.k_feature_names_)
print("\n✅ Características seleccionadas:")
print(f"Índices: {sfs_f.k_feature_idx_}")
print(f"Nombres: {selected_features_f}")
x_selected_train = x_train[selected_features_f]
x_selected_test = x_test[selected_features_f]
Features: 43/43
✅ Características seleccionadas: Índices: (0, 1, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42) Nombres: ['number_of_elements', 'mean_atomic_mass', 'gmean_atomic_mass', 'wtd_entropy_atomic_mass', 'range_atomic_mass', 'wtd_range_atomic_mass', 'mean_fie', 'wtd_mean_fie', 'wtd_entropy_fie', 'mean_atomic_radius', 'gmean_atomic_radius', 'wtd_range_atomic_radius', 'mean_Density', 'wtd_entropy_Density', 'range_Density', 'wtd_range_Density', 'mean_ElectronAffinity', 'wtd_mean_ElectronAffinity', 'wtd_gmean_ElectronAffinity', 'entropy_ElectronAffinity', 'wtd_entropy_ElectronAffinity', 'range_ElectronAffinity', 'wtd_range_ElectronAffinity', 'mean_FusionHeat', 'wtd_entropy_FusionHeat', 'range_FusionHeat', 'wtd_range_FusionHeat', 'mean_ThermalConductivity', 'wtd_mean_ThermalConductivity', 'gmean_ThermalConductivity', 'wtd_gmean_ThermalConductivity', 'entropy_ThermalConductivity', 'wtd_entropy_ThermalConductivity', 'range_ThermalConductivity', 'mean_Valence', 'range_Valence', 'wtd_range_Valence', 'wtd_std_Valence']
Se utilizó el método de selección hacia adelante debido a que se busca un modelo con el menor número de variables posibles para que tenga un mejor rendimiento. El número de variables disminuyó, lo cual hará el entrenamiento del modelo más fácil de llevar a nivel computacional y también lo hace más interpretable.
Modelo de Regresión Lineal¶
Primero se genera un modelo de regresión lineal utilizando la función LinearRegression de sklearn.linear_model.
Modifiqué las variables de entrada para que solo tomaran en cuenta las variables seleccionadas tras el proceso de selección de características. Utilizando la función fit() se entreno el modelo con los datos de entrenamiento y después con la función predict() se probó el modelo con los datos de prueba. También se uso la función r2_score y se calculo la R^2 del modelo para conocer su desempeño. La gráfica muestra el desempeño del modelo contra los valores reales.
In [18]:
model.fit(x_selected_train, y_train)
y_pred = model.predict(x_selected_test)
r2_linear = r2_score(y_test,y_pred)
plt.scatter(y_test, y_pred)
plt.xlabel("Temperatura Crítica Real")
plt.ylabel("Temperatura Crítica Estimada")
print(f"\n📊 R^2 en datos de prueba: {r2_linear:.4f}")
📊 R^2 en datos de prueba: 0.6561
Modelo de Regresión Polinomial¶
Ahora se probará un modelo de regresión polinomial. Para este se usa la función PolynomialFeatures de sklearn.preprocessing. Se usa un grado de 2 para equilibrar entre eficiencia computacional y buena calidad. Se transforman los datos con la función fit_transform y de nuevo utilizando la función LinearRegression se entrena el modelo polinomial. Una vez entrenado, se prueba con los datos de validación y se calcula su R^2. La gráfica muestra el desempeño del modelo contra los valores reales.
In [42]:
from sklearn.preprocessing import PolynomialFeatures
poly = PolynomialFeatures(degree=2)
x_poly_train = poly.fit_transform(x_selected_train)
x_poly_test = poly.fit_transform(x_selected_test)
model = LinearRegression()
model.fit(x_poly_train, y_train)
yhat_poly = model.predict(x_poly_test)
r2_poly = r2_score(y_test,yhat_poly)
print("R^2:",r2_poly)
plt.scatter(y_test,yhat_poly, color = "red",alpha = 0.5, label="Regresión Polinomial")
plt.title("Regresión Polinomial")
plt.xlabel("Temperatura Crítica Real")
plt.ylabel("Temperatura Crítica Estimada")
plt.legend()
plt.grid(True)
plt.show()
R^2: 0.7827518846207967
Métricas de Error y Linealidad¶
Para comparar ambos modelos se calcula el mean square error (MSE) y la R^2 de ambos modelos de acuerdo con el desempeño de sus predicciones en los datos de validación.
In [43]:
from sklearn.metrics import mean_squared_error
mse_poly = mean_squared_error(y_test,yhat_poly)
mse_linear = mean_squared_error(y_test,y_pred)
print("MSE de Regresión Lineal:",mse_linear)
print("R^2 de Regresión Lineal:",r2_linear)
print("MSE de Regresión Polinomial:",mse_poly)
print("R^2 de Regresión Polinomial:",r2_poly)
MSE de Regresión Lineal: 407.421780417495 R^2 de Regresión Lineal: 0.656058819903386 MSE de Regresión Polinomial: 257.3452063381222 R^2 de Regresión Polinomial: 0.7827518846207967
Tras comparar ambos modelos, el modelo de regresión polinomial es el claro ganador ya que tiene un menor error (MSE) y un mayor nivel de linealidad (R^2) por lo tanto será el modelo seleccionado.
Inferencia con el Modelo¶
Tras obtener el modelo ganador, se utiliza la librería statsmodels.api para obtener información relevante para generar inferencias a partir del modelo.
In [44]:
import statsmodels.api as sm
poly_inf = sm.OLS(y_train,x_poly_train).fit()
print(poly_inf.summary())
OLS Regression Results
==============================================================================
Dep. Variable: critical_temp R-squared: 0.831
Model: OLS Adj. R-squared: 0.818
Method: Least Squares F-statistic: 64.09
Date: Thu, 20 Feb 2025 Prob (F-statistic): 0.00
Time: 06:36:51 Log-Likelihood: -44811.
No. Observations: 10966 AIC: 9.118e+04
Df Residuals: 10186 BIC: 9.688e+04
Df Model: 779
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const -516.0799 370.659 -1.392 0.164 -1242.645 210.485
x1 -10.9082 68.357 -0.160 0.873 -144.901 123.084
x2 8.7944 4.549 1.933 0.053 -0.123 17.712
x3 -10.7145 4.370 -2.452 0.014 -19.280 -2.149
x4 529.8335 278.452 1.903 0.057 -15.987 1075.654
x5 -2.6347 0.909 -2.897 0.004 -4.417 -0.852
x6 3.6937 1.848 1.998 0.046 0.070 7.317
x7 1.0021 0.762 1.316 0.188 -0.491 2.495
x8 -0.0353 0.617 -0.057 0.954 -1.244 1.174
x9 -218.0254 270.618 -0.806 0.420 -748.489 312.438
x10 1.3011 6.036 0.216 0.829 -10.531 13.133
x11 1.4149 6.892 0.205 0.837 -12.094 14.924
x12 1.6817 1.036 1.623 0.105 -0.350 3.713
x13 -0.0010 0.025 -0.040 0.968 -0.050 0.048
x14 603.6378 269.446 2.240 0.025 75.470 1131.805
x15 0.0003 0.010 0.034 0.973 -0.020 0.020
x16 0.0116 0.024 0.479 0.632 -0.036 0.059
x17 1.1129 2.385 0.467 0.641 -3.563 5.789
x18 4.8201 4.902 0.983 0.325 -4.789 14.429
x19 -0.9670 3.674 -0.263 0.792 -8.169 6.235
x20 -106.9873 203.615 -0.525 0.599 -506.112 292.137
x21 -394.9268 233.534 -1.691 0.091 -852.699 62.845
x22 -1.0258 1.038 -0.988 0.323 -3.061 1.010
x23 -3.4095 2.430 -1.403 0.161 -8.173 1.354
x24 5.2595 6.556 0.802 0.422 -7.591 18.110
x25 -506.4936 201.017 -2.520 0.012 -900.527 -112.461
x26 0.8163 2.867 0.285 0.776 -4.803 6.435
x27 -7.1118 6.839 -1.040 0.298 -20.518 6.295
x28 0.9293 2.284 0.407 0.684 -3.548 5.407
x29 4.7342 1.598 2.963 0.003 1.602 7.866
x30 -1.7906 2.362 -0.758 0.448 -6.420 2.839
x31 -3.4320 2.064 -1.663 0.096 -7.478 0.614
x32 -205.2880 181.079 -1.134 0.257 -560.239 149.663
x33 226.7734 136.363 1.663 0.096 -40.525 494.072
x34 -0.5566 0.761 -0.731 0.465 -2.049 0.935
x35 -52.0638 45.572 -1.142 0.253 -141.394 37.267
x36 9.4912 35.094 0.270 0.787 -59.299 78.282
x37 -50.3646 37.573 -1.340 0.180 -124.015 23.286
x38 43.8329 91.948 0.477 0.634 -136.403 224.069
x39 -5.7850 3.710 -1.559 0.119 -13.057 1.486
x40 0.7475 0.395 1.892 0.059 -0.027 1.522
x41 -0.4907 0.410 -1.198 0.231 -1.294 0.313
x42 4.5227 26.066 0.174 0.862 -46.573 55.618
x43 -0.1069 0.078 -1.377 0.168 -0.259 0.045
x44 -0.2229 0.237 -0.942 0.346 -0.687 0.241
x45 0.1295 0.067 1.936 0.053 -0.002 0.261
x46 -0.0643 0.061 -1.061 0.288 -0.183 0.054
x47 -64.0233 23.791 -2.691 0.007 -110.657 -17.389
x48 -2.2702 0.595 -3.816 0.000 -3.436 -1.104
x49 2.3054 0.688 3.352 0.001 0.957 3.653
x50 0.1022 0.123 0.833 0.405 -0.138 0.343
x51 -0.0052 0.003 -1.809 0.071 -0.011 0.000
x52 6.5109 24.746 0.263 0.792 -41.996 55.017
x53 -0.0006 0.001 -0.586 0.558 -0.002 0.001
x54 0.0060 0.003 1.894 0.058 -0.000 0.012
x55 -0.1460 0.242 -0.602 0.547 -0.621 0.329
x56 0.4647 0.441 1.054 0.292 -0.399 1.329
x57 -0.5527 0.311 -1.777 0.076 -1.162 0.057
x58 39.8594 21.383 1.864 0.062 -2.056 81.774
x59 60.1410 24.791 2.426 0.015 11.545 108.737
x60 0.1254 0.098 1.285 0.199 -0.066 0.317
x61 0.1307 0.300 0.436 0.663 -0.457 0.718
x62 -0.6929 0.881 -0.786 0.432 -2.421 1.035
x63 -9.3110 19.808 -0.470 0.638 -48.139 29.517
x64 0.7284 0.243 2.995 0.003 0.252 1.205
x65 -0.4463 0.912 -0.489 0.625 -2.234 1.341
x66 0.0820 0.210 0.390 0.697 -0.330 0.494
x67 0.2060 0.145 1.423 0.155 -0.078 0.490
x68 -0.2679 0.254 -1.054 0.292 -0.766 0.230
x69 -0.2632 0.207 -1.268 0.205 -0.670 0.144
x70 3.6663 18.705 0.196 0.845 -32.999 40.331
x71 21.7815 11.647 1.870 0.061 -1.049 44.612
x72 0.0752 0.075 1.002 0.317 -0.072 0.222
x73 -15.4993 5.060 -3.063 0.002 -25.417 -5.582
x74 -2.3348 2.725 -0.857 0.392 -7.676 3.007
x75 7.3668 4.812 1.531 0.126 -2.067 16.800
x76 0.2018 7.571 0.027 0.979 -14.638 15.041
x77 0.0337 0.019 1.760 0.078 -0.004 0.071
x78 -0.0659 0.034 -1.944 0.052 -0.132 0.001
x79 0.9365 1.792 0.522 0.601 -2.577 4.450
x80 0.0134 0.007 1.788 0.074 -0.001 0.028
x81 -0.0725 0.017 -4.332 0.000 -0.105 -0.040
x82 0.0090 0.004 2.017 0.044 0.000 0.018
x83 -0.0149 0.004 -4.131 0.000 -0.022 -0.008
x84 -8.3554 1.682 -4.969 0.000 -11.651 -5.059
x85 -0.1013 0.037 -2.770 0.006 -0.173 -0.030
x86 0.1018 0.042 2.433 0.015 0.020 0.184
x87 -0.0031 0.008 -0.391 0.696 -0.018 0.012
x88 -0.0004 0.000 -1.539 0.124 -0.001 9.88e-05
x89 -0.1823 1.815 -0.100 0.920 -3.740 3.375
x90 -0.0002 8.48e-05 -1.914 0.056 -0.000 3.88e-06
x91 0.0012 0.000 5.985 0.000 0.001 0.002
x92 0.0085 0.016 0.518 0.605 -0.024 0.041
x93 0.1065 0.030 3.506 0.000 0.047 0.166
x94 -0.0825 0.023 -3.641 0.000 -0.127 -0.038
x95 1.8145 1.420 1.278 0.201 -0.968 4.597
x96 1.9861 1.729 1.149 0.251 -1.403 5.375
x97 -0.0042 0.007 -0.631 0.528 -0.017 0.009
x98 -0.0342 0.018 -1.904 0.057 -0.069 0.001
x99 0.0094 0.058 0.161 0.872 -0.105 0.123
x100 -0.1742 1.347 -0.129 0.897 -2.815 2.467
x101 -0.0446 0.020 -2.242 0.025 -0.084 -0.006
x102 -0.0156 0.060 -0.257 0.797 -0.134 0.103
x103 0.0313 0.016 1.976 0.048 0.000 0.062
x104 -0.0052 0.009 -0.568 0.570 -0.023 0.013
x105 -0.0190 0.018 -1.080 0.280 -0.053 0.015
x106 -0.0091 0.014 -0.647 0.518 -0.037 0.019
x107 0.1508 1.233 0.122 0.903 -2.265 2.567
x108 0.8221 0.840 0.979 0.328 -0.824 2.468
x109 -0.0075 0.005 -1.435 0.151 -0.018 0.003
x110 -0.2029 0.350 -0.580 0.562 -0.888 0.482
x111 0.2133 0.191 1.116 0.265 -0.161 0.588
x112 -0.6984 0.322 -2.171 0.030 -1.329 -0.068
x113 -1.1536 0.533 -2.166 0.030 -2.198 -0.110
x114 0.0190 0.017 1.120 0.263 -0.014 0.052
x115 1.0663 1.839 0.580 0.562 -2.538 4.670
x116 -0.0076 0.007 -1.088 0.277 -0.021 0.006
x117 0.0706 0.017 4.267 0.000 0.038 0.103
x118 -0.0054 0.004 -1.256 0.209 -0.014 0.003
x119 0.0112 0.003 3.194 0.001 0.004 0.018
x120 7.7507 1.690 4.586 0.000 4.438 11.063
x121 0.0823 0.036 2.280 0.023 0.012 0.153
x122 -0.0627 0.041 -1.526 0.127 -0.143 0.018
x123 -0.0060 0.008 -0.740 0.459 -0.022 0.010
x124 0.0007 0.000 2.885 0.004 0.000 0.001
x125 -2.9379 1.846 -1.592 0.111 -6.556 0.680
x126 7.221e-05 8.04e-05 0.898 0.369 -8.54e-05 0.000
x127 -0.0012 0.000 -6.104 0.000 -0.002 -0.001
x128 0.0056 0.017 0.340 0.734 -0.027 0.038
x129 -0.1110 0.032 -3.454 0.001 -0.174 -0.048
x130 0.0846 0.024 3.515 0.000 0.037 0.132
x131 -2.0461 1.491 -1.372 0.170 -4.969 0.876
x132 -1.7892 1.771 -1.011 0.312 -5.260 1.681
x133 0.0019 0.007 0.280 0.779 -0.011 0.015
x134 0.0328 0.018 1.820 0.069 -0.003 0.068
x135 -0.0791 0.054 -1.465 0.143 -0.185 0.027
x136 1.0860 1.388 0.783 0.434 -1.634 3.806
x137 0.0445 0.019 2.340 0.019 0.007 0.082
x138 0.0460 0.057 0.802 0.423 -0.066 0.158
x139 -0.0538 0.017 -3.224 0.001 -0.087 -0.021
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x770 0.5061 2.065 0.245 0.806 -3.542 4.554
x771 -2.8247 2.585 -1.093 0.275 -7.892 2.243
x772 -0.6597 2.676 -0.247 0.805 -5.905 4.586
x773 15.9488 6.562 2.431 0.015 3.087 28.811
x774 1.3191 0.901 1.464 0.143 -0.447 3.085
x775 0.9776 2.501 0.391 0.696 -3.924 5.879
x776 -7.3609 4.636 -1.588 0.112 -16.448 1.726
x777 1.1157 1.893 0.589 0.556 -2.595 4.826
x778 -12.9081 6.263 -2.061 0.039 -25.184 -0.632
x779 32.6915 7.728 4.230 0.000 17.543 47.840
==============================================================================
Omnibus: 1429.179 Durbin-Watson: 2.005
Prob(Omnibus): 0.000 Jarque-Bera (JB): 3371.878
Skew: -0.763 Prob(JB): 0.00
Kurtosis: 5.247 Cond. No. 3.16e+11
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 3.16e+11. This might indicate that there are
strong multicollinearity or other numerical problems.
Al ser un modelo polinomial de segundo grado basado en 38 variables, es difícil interpretar a la perfección cada uno de los coeficientes, ya que son un total de 779. Sin embargo se pueden realizar algunas inferencias.
Por ejemplo:
- El término x^5 reduce la temperatura crítica de un superconductor entre -4.417 y -0.85 grados con un 95% de confianza.
- El término x^117 aumenta la temperatura crítica de un superconductor entre 0.038 y 0.103 grados con un 95% de confianza.
- El término x^424 aumenta la tempratura crítica de un superconductor entre 0.003 y 0.008 grados con un 95% de confianza.
En general, utilizando estos datos y el modelo, se pueden obtener conclusiones del efecto de las variables en la temperatura crítica. Aún así, no se puede saber el efecto específico de cada variable, ya que al tener varias y trabajar con una regresión polinomial de grado 2, cada término representa no solo el cuadrado de cada variable si no también las interacciones entre cada par de variables posibles.
Con esto en mente, se puede concluir que el modelo es bueno para realizar predicciones, sin embargo, para realizar inferencias, sería mejor considerar modelos más interpretables como la regresión lineal.